10-2 Special mapping equations 277
10-22 Conformal mapping (Mercator projection)
The requirement for conformality leads to the postulate (10.9). Again, the integration constant is
determined from the additional constraint that for Φ = 0 the coordinate y should vanish, namely
y = 0 ⇒ const. = 0. Therefore, the mapping equations are provided by (10.10). The left principal
stretches are provided by (10.11). The parallel circle Φ = ±Φ 0 is mapped free from any distortion.
Compare with Fig. 10.5.
Λ 1 = Λ 2
⇒
cos Φ 0
cos Φ
=
1
R
df
dΦ
⇒ df = R
cos Φ 0 dΦ
cos Φ
⇒
df =f (Φ) = R cos Φ 0
dΦ
cos Φ
= R cos Φ 0 ln cot
π
4
−
Φ
2
+ const. ,
(10.9)
x
y
= R cos Φ 0
Λ
ln cot
π
4 −
Φ
2
= R cos Φ 0
Λ
ln tan
π
4 +
Φ
2
,
(10.10)
Λ 1 = Λ 2 =
cos Φ 0
cos Φ
.
(10.11)
Fig. 10.5. Mapping the sphere to a cylinder: polar aspect, conformal mapping, Φ 0 = 0
◦ : tangent cylinder
(Mercator projection).